#72209677

order	2916
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting

C₃⁵ ⋊ A₄

Subgroups

	subgroup	G/H
center, Frattini	C₃	C₃×#1113⋊C₃
commutator, cosocle	C₃×#1113	C₃²
Fitting	C₃⁵	A₄
socle	C₃⁴	C₃×A₄
hypercenter	C₃²	#1113⋊C₃
hypocenter	#1113	H(3,𝔽₃)
LCS	G ▷ C₃×#1113 ▷ #1113
UCS	C₁ ◁ C₃ ◁ C₃²

554 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

3^3,2^2,3^3|b^a=bc,d^b=e,e^b=de,f^b=g,f^d=f^2,g^b=h,g^d=g^2,g^e=g^2,h^b=f,h^e=h^2

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209678
maximal quotient of	#72209947

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [3, 3, 3, 2, 2, 3, 3, 3]);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.6);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);